This document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and total biomass as the focal ecosystem function.

Important terms:

  • Stage: With seed rain, without seed rain
  • Ninitial: Planted species richness

1 Grass1

Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.

1.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                2.44      3.60    -4.41     9.57 1.00     2169     2035
## StageWithoutseedrain           -12.24      4.04   -20.11    -4.50 1.00     2034     1838
## StageWithseedrain:Shannon       17.72      1.50    14.80    20.59 1.00     2148     2128
## StageWithoutseedrain:Shannon    25.71      1.98    21.89    29.56 1.00     2052     1932
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    23.18      0.60    22.03    24.40 1.00     2921     2567
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2959759 0.02351531 0.2491463 0.3411242

1.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

1.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

1.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

1.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


1.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                83.58     26.32    32.36   135.89 1.00     5827     2749
## Ninitial4:StageWithseedrain               150.99     24.64   102.15   198.74 1.00     6218     2977
## Ninitial8:StageWithseedrain               167.37     19.75   128.31   206.30 1.00     4548     2418
## Ninitial16:StageWithseedrain              250.82     17.19   216.15   284.27 1.00     5185     2576
## Ninitial32:StageWithseedrain              278.51     20.11   237.99   316.43 1.00     5223     2962
## Ninitial2:StageWithoutseedrain             23.04     10.35     2.65    43.66 1.00     5084     2871
## Ninitial4:StageWithoutseedrain             65.17     15.54    34.07    95.94 1.00     5889     2425
## Ninitial8:StageWithoutseedrain             89.01     14.25    60.70   116.00 1.00     5403     2907
## Ninitial16:StageWithoutseedrain           184.66     17.79   150.14   219.32 1.00     6024     2766
## Ninitial32:StageWithoutseedrain           188.79     23.28   142.48   233.99 1.00     5615     2702
## Ninitial2:StageWithseedrain:Shannon       -36.98     16.37   -69.52    -4.99 1.00     5759     2780
## Ninitial4:StageWithseedrain:Shannon       -54.03     11.30   -76.05   -32.21 1.00     6206     3023
## Ninitial8:StageWithseedrain:Shannon       -47.74      7.43   -62.38   -33.01 1.00     4704     2553
## Ninitial16:StageWithseedrain:Shannon      -64.99      5.84   -76.29   -53.37 1.00     5168     2831
## Ninitial32:StageWithseedrain:Shannon      -64.73      6.36   -76.81   -51.98 1.00     5074     3036
## Ninitial2:StageWithoutseedrain:Shannon     -3.34      6.97   -17.16    10.41 1.00     5037     2791
## Ninitial4:StageWithoutseedrain:Shannon    -19.61      7.96   -35.27    -3.88 1.00     5917     2460
## Ninitial8:StageWithoutseedrain:Shannon    -23.86      6.33   -35.99   -11.16 1.00     5436     2823
## Ninitial16:StageWithoutseedrain:Shannon   -52.74      7.33   -67.11   -38.37 1.00     6061     2596
## Ninitial32:StageWithoutseedrain:Shannon   -44.01      8.89   -61.21   -26.23 1.00     5570     2822
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    15.05      0.43    14.25    15.93 1.00     6366     2702
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##    Estimate  Est.Error      Q2.5     Q97.5
## R2 0.699607 0.01080048 0.6768392 0.7193672

1.2.1 Posterior predictive checks

1.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

1.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

1.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


2 Grass2

Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.

2.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               44.49      1.56    41.45    47.59 1.00     2100     2157
## StageWithoutseedrain            37.55      1.79    33.98    41.03 1.00     2021     2313
## StageWithseedrain:Shannon        8.72      0.56     7.67     9.81 1.00     2048     2289
## StageWithoutseedrain:Shannon    15.06      0.79    13.50    16.60 1.00     2015     2289
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    12.23      0.31    11.64    12.89 1.00     3099     2284
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4411094 0.01965722 0.4009131 0.4781665

2.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

2.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

2.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

2.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


2.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                85.27     33.54    17.89   150.48 1.00     5967     2907
## Ninitial4:StageWithseedrain               136.35     35.73    65.21   206.05 1.00     5692     2835
## Ninitial8:StageWithseedrain               132.92     32.44    69.06   197.56 1.00     5874     2851
## Ninitial16:StageWithseedrain               92.40     50.17    -4.24   187.40 1.00     4487     2735
## Ninitial32:StageWithseedrain               43.79     61.62   -78.41   164.43 1.00     5436     2580
## Ninitial2:StageWithoutseedrain            -16.74     22.64   -60.98    27.64 1.00     5015     2750
## Ninitial4:StageWithoutseedrain             27.61     13.99     0.00    55.05 1.00     5448     2253
## Ninitial8:StageWithoutseedrain             38.52     10.67    17.45    59.66 1.00     5528     2653
## Ninitial16:StageWithoutseedrain            47.39     11.77    23.89    70.57 1.00     4933     2519
## Ninitial32:StageWithoutseedrain            72.57     14.87    42.50   102.05 1.00     5099     2890
## Ninitial2:StageWithseedrain:Shannon       -16.30     20.34   -55.82    24.25 1.00     5960     2817
## Ninitial4:StageWithseedrain:Shannon       -29.94     15.50   -59.98     1.16 1.00     5683     2813
## Ninitial8:StageWithseedrain:Shannon       -20.03     11.20   -42.51     1.89 1.00     5866     2744
## Ninitial16:StageWithseedrain:Shannon       -4.68     14.15   -31.68    22.59 1.00     4519     2818
## Ninitial32:StageWithseedrain:Shannon        7.92     14.68   -20.80    36.92 1.00     5437     2672
## Ninitial2:StageWithoutseedrain:Shannon     46.23     13.98    18.61    73.66 1.00     5009     2680
## Ninitial4:StageWithoutseedrain:Shannon     21.86      7.61     7.20    36.94 1.00     5457     2265
## Ninitial8:StageWithoutseedrain:Shannon     17.78      4.81     8.03    27.23 1.00     5501     2690
## Ninitial16:StageWithoutseedrain:Shannon    12.24      4.32     3.76    20.91 1.00     4942     2554
## Ninitial32:StageWithoutseedrain:Shannon     3.26      4.43    -5.63    12.29 1.00     5087     2932
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.05      0.27     8.55     9.61 1.00     6294     3028
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##    Estimate  Est.Error      Q2.5     Q97.5
## R2 0.494272 0.01979012 0.4522059 0.5306177

2.2.1 Posterior predictive checks

2.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

2.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

2.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


3 Grass3

May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.

3.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               46.13      1.37    43.44    48.76 1.00     1800     2269
## StageWithoutseedrain            46.10      1.49    43.10    49.01 1.00     2113     2319
## StageWithseedrain:Shannon        2.67      0.51     1.70     3.67 1.00     1729     2041
## StageWithoutseedrain:Shannon     2.84      0.59     1.69     4.02 1.00     2190     2300
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.96      0.25     9.49    10.45 1.00     3254     2501
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5      Q97.5
## R2 0.06486826 0.01628797 0.03579288 0.09836984

3.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

3.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

3.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


3.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                18.84     38.76   -57.94    94.43 1.00     3971     3032
## Ninitial4:StageWithseedrain                -7.75     26.33   -59.09    43.28 1.00     3674     2731
## Ninitial8:StageWithseedrain                 3.74     26.30   -46.91    55.59 1.00     3700     2938
## Ninitial16:StageWithseedrain               58.09     22.36    13.72   101.67 1.00     4016     2670
## Ninitial32:StageWithseedrain               47.95     31.43   -12.66   108.67 1.00     3796     2821
## Ninitial2:StageWithoutseedrain             64.74      6.51    51.80    77.28 1.00     3622     2826
## Ninitial4:StageWithoutseedrain             25.58     11.03     3.94    46.83 1.00     3971     2806
## Ninitial8:StageWithoutseedrain             36.99     11.79    13.29    60.15 1.00     3568     2554
## Ninitial16:StageWithoutseedrain            64.85     21.20    24.19   106.44 1.00     3837     3038
## Ninitial32:StageWithoutseedrain            79.01     32.15    16.41   142.19 1.00     4088     2839
## Ninitial2:StageWithseedrain:Shannon        18.04     23.23   -27.30    64.11 1.00     3971     3042
## Ninitial4:StageWithseedrain:Shannon        25.42     11.53     3.05    47.68 1.00     3683     2664
## Ninitial8:StageWithseedrain:Shannon        17.48      9.30    -0.66    35.58 1.00     3705     2970
## Ninitial16:StageWithseedrain:Shannon       -0.84      6.63   -13.64    12.36 1.00     4030     2792
## Ninitial32:StageWithseedrain:Shannon        2.58      8.26   -13.28    18.71 1.00     3795     2955
## Ninitial2:StageWithoutseedrain:Shannon     -9.89      4.04   -17.72    -2.10 1.00     3692     2765
## Ninitial4:StageWithoutseedrain:Shannon     11.56      5.14     1.60    21.52 1.00     3954     2822
## Ninitial8:StageWithoutseedrain:Shannon      6.12      4.49    -2.67    15.14 1.00     3542     2568
## Ninitial16:StageWithoutseedrain:Shannon    -3.40      6.67   -16.46     9.46 1.00     3835     3063
## Ninitial32:StageWithoutseedrain:Shannon    -5.79      9.13   -23.72    12.00 1.00     4090     2786
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     6.73      0.19     6.38     7.11 1.00     5931     2917
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error     Q2.5     Q97.5
## R2 0.2337697 0.02421294 0.186099 0.2806291

3.2.1 Posterior predictive checks

3.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

3.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

3.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


4 Forest1

Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.

4.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               28.07      3.28    21.72    34.60 1.00     2013     2067
## StageWithoutseedrain            30.72      3.81    23.18    38.25 1.00     2158     1845
## StageWithseedrain:Shannon       15.02      1.96    11.09    18.75 1.00     1938     2034
## StageWithoutseedrain:Shannon    13.84      2.80     8.31    19.27 1.00     2093     1872
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    21.88      0.53    20.89    22.93 1.00     2550     2379
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error       Q2.5     Q97.5
## R2 0.1041526 0.01951819 0.06759033 0.1425655

4.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


4.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                66.92     11.66    43.47    89.73 1.00     3650     3008
## Ninitial4:StageWithseedrain                89.20      9.86    69.92   108.30 1.00     4907     2476
## Ninitial8:StageWithseedrain                94.26     11.31    71.39   117.04 1.00     4220     2718
## Ninitial16:StageWithseedrain               78.08     14.14    50.81   105.56 1.00     4356     3168
## Ninitial32:StageWithseedrain               41.33     20.33     1.41    80.74 1.00     4013     2704
## Ninitial2:StageWithoutseedrain             65.65     14.48    38.02    94.47 1.00     4459     2669
## Ninitial4:StageWithoutseedrain             75.20      8.97    57.52    93.36 1.00     4337     2669
## Ninitial8:StageWithoutseedrain            100.70      9.79    81.61   119.80 1.00     4512     2932
## Ninitial16:StageWithoutseedrain            58.36      9.52    39.93    77.28 1.00     4300     2634
## Ninitial32:StageWithoutseedrain            46.68      8.74    29.37    63.71 1.00     4358     3062
## Ninitial2:StageWithseedrain:Shannon       -24.44      9.77   -43.55    -5.02 1.00     3678     2805
## Ninitial4:StageWithseedrain:Shannon       -28.71      7.39   -42.96   -14.02 1.00     4871     2180
## Ninitial8:StageWithseedrain:Shannon       -22.95      7.26   -37.50    -8.07 1.00     4152     2894
## Ninitial16:StageWithseedrain:Shannon       -6.92      7.11   -20.58     6.81 1.00     4396     3146
## Ninitial32:StageWithseedrain:Shannon       10.07      8.23    -6.00    26.23 1.00     4051     2844
## Ninitial2:StageWithoutseedrain:Shannon    -27.94     13.49   -54.69    -2.37 1.00     4467     2770
## Ninitial4:StageWithoutseedrain:Shannon    -22.75      7.38   -37.69    -7.93 1.00     4394     2594
## Ninitial8:StageWithoutseedrain:Shannon    -33.87      7.33   -48.17   -19.68 1.00     4484     3051
## Ninitial16:StageWithoutseedrain:Shannon     1.76      6.13   -10.56    13.57 1.00     4291     2644
## Ninitial32:StageWithoutseedrain:Shannon     8.29      4.88    -1.26    18.04 1.00     4361     2787
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    17.65      0.50    16.72    18.66 1.01     8326     2825
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.3175321 0.02487158 0.2664479 0.3639134

4.2.1 Posterior predictive checks

4.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


5 Forest2

Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.

5.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               31.86      1.40    29.12    34.62 1.00     2025     1927
## StageWithoutseedrain            28.81      1.72    25.50    32.35 1.00     1814     1787
## StageWithseedrain:Shannon       -0.42      0.53    -1.45     0.59 1.00     1954     1721
## StageWithoutseedrain:Shannon    -6.13      0.94    -8.05    -4.29 1.00     1785     1837
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    10.62      0.27    10.11    11.19 1.00     3144     2389
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2944755 0.02322893 0.2483561 0.3394765

5.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

5.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

5.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

5.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


5.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                80.24     10.61    59.37   101.22 1.00     4164     2950
## Ninitial4:StageWithseedrain               101.47     12.73    76.89   127.03 1.00     3905     3046
## Ninitial8:StageWithseedrain                70.65     17.85    35.35   105.04 1.00     3966     3022
## Ninitial16:StageWithseedrain               84.78     30.22    27.69   143.42 1.00     4518     3175
## Ninitial32:StageWithseedrain              116.16     44.15    30.63   201.42 1.00     3862     2871
## Ninitial2:StageWithoutseedrain             14.02      5.80     2.79    25.55 1.00     4454     2848
## Ninitial4:StageWithoutseedrain              3.94      5.44    -6.63    14.58 1.00     4341     3158
## Ninitial8:StageWithoutseedrain             15.49      6.16     3.28    27.85 1.00     3999     2822
## Ninitial16:StageWithoutseedrain             8.31      5.57    -2.23    19.41 1.00     3210     2470
## Ninitial32:StageWithoutseedrain            12.79      7.66    -2.46    27.86 1.00     4103     2956
## Ninitial2:StageWithseedrain:Shannon       -31.66      6.98   -45.28   -17.95 1.00     4156     3160
## Ninitial4:StageWithseedrain:Shannon       -33.59      6.02   -45.66   -21.88 1.00     3905     2985
## Ninitial8:StageWithseedrain:Shannon       -14.83      6.66   -27.50    -1.80 1.00     3983     3030
## Ninitial16:StageWithseedrain:Shannon      -16.25      9.03   -33.79     0.87 1.00     4517     3187
## Ninitial32:StageWithseedrain:Shannon      -21.66     11.23   -43.46     0.09 1.00     3870     2774
## Ninitial2:StageWithoutseedrain:Shannon      7.47      4.29    -1.08    15.86 1.00     4444     2869
## Ninitial4:StageWithoutseedrain:Shannon      9.14      3.38     2.58    15.72 1.00     4311     3002
## Ninitial8:StageWithoutseedrain:Shannon      0.01      3.09    -6.03     6.13 1.00     4163     2767
## Ninitial16:StageWithoutseedrain:Shannon     2.38      2.61    -2.79     7.41 1.00     3064     2933
## Ninitial32:StageWithoutseedrain:Shannon    -0.03      3.08    -6.04     6.06 1.00     4020     2602
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     8.46      0.23     8.00     8.93 1.00     7049     3089
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5    Q97.5
## R2 0.4971028 0.01972711 0.4556248 0.533383

5.2.1 Posterior predictive checks

5.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

5.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

5.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


6 Dryland

Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.

6.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 770) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               73.93      1.28    71.42    76.43 1.00     2031     2143
## StageWithoutseedrain            77.93      1.66    74.74    81.21 1.00     2042     2209
## StageWithseedrain:Shannon        1.17      0.54     0.14     2.21 1.00     2048     2236
## StageWithoutseedrain:Shannon    -1.54      1.05    -3.66     0.49 1.00     2066     2159
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.38      0.25     8.92     9.87 1.01     3041     2389
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate   Est.Error        Q2.5      Q97.5
## R2 0.01443426 0.007817357 0.002762788 0.03248531

6.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

6.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

6.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

6.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.


6.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: biomass ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                63.10      6.65    49.61    76.15 1.00     3401     2635
## Ninitial4:StageWithseedrain                63.05      8.83    46.19    80.79 1.00     3237     2869
## Ninitial8:StageWithseedrain                44.71     11.09    22.75    66.31 1.00     3394     2547
## Ninitial16:StageWithseedrain               42.54     16.03    11.38    74.03 1.00     3825     3004
## Ninitial32:StageWithseedrain                1.27     20.78   -39.27    41.63 1.00     3640     2943
## Ninitial2:StageWithoutseedrain             73.27      4.34    64.79    81.89 1.00     3788     2946
## Ninitial4:StageWithoutseedrain             60.23      4.41    51.65    68.91 1.00     3322     2524
## Ninitial8:StageWithoutseedrain             55.04      4.62    45.93    64.24 1.00     3935     3062
## Ninitial16:StageWithoutseedrain            41.82      6.64    28.58    54.74 1.00     3537     2754
## Ninitial32:StageWithoutseedrain            38.28      9.57    18.90    57.06 1.00     3335     2585
## Ninitial2:StageWithseedrain:Shannon         9.31      4.57     0.34    18.51 1.00     3379     2434
## Ninitial4:StageWithseedrain:Shannon         7.23      4.62    -2.21    15.99 1.00     3219     2914
## Ninitial8:StageWithseedrain:Shannon        13.35      4.58     4.39    22.36 1.00     3376     2708
## Ninitial16:StageWithseedrain:Shannon       11.51      5.42     0.87    22.06 1.00     3828     3014
## Ninitial32:StageWithseedrain:Shannon       21.98      5.99    10.32    33.72 1.00     3637     2925
## Ninitial2:StageWithoutseedrain:Shannon      5.46      3.44    -1.43    12.24 1.00     3819     2942
## Ninitial4:StageWithoutseedrain:Shannon     11.38      3.02     5.34    17.29 1.00     3515     2812
## Ninitial8:StageWithoutseedrain:Shannon     11.96      2.76     6.42    17.32 1.00     3961     3074
## Ninitial16:StageWithoutseedrain:Shannon    16.42      3.56     9.62    23.50 1.00     3537     2676
## Ninitial32:StageWithoutseedrain:Shannon    16.34      4.75     7.01    25.99 1.00     3278     2564
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     7.95      0.23     7.50     8.43 1.00     5706     2674
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##    Estimate  Est.Error      Q2.5     Q97.5
## R2 0.224007 0.02505269 0.1746218 0.2745683

6.2.1 Posterior predictive checks

6.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

6.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

6.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.